Maximal implies prime

This article gives the statement (and possibly proof) of an implication between properties of ideals in commutative unital rings; viz., any ideal in a commutative unital ring satisfying the first property, also satisfies the second

Statement

Any maximal ideal in a commutative unital ring is a prime ideal.

Definitions used

Maximal ideal

Further information: Maximal ideal

Prime ideal

Further information: Prime ideal

Proof

As quotient-determined properties

One proof uses the characterization of maximal and prime ideals in terms of their quotients, namely:

• An ideal is maximal iff the quotient ring is a field
• An ideal is prime iff the quotient ring is an integral domain

We know that the property of commutative unital rings of being a field is stronger than the property of being an integral domain, hence any maximal ideal is prime.

Hands-on proof

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